Improved Multi-Pass Streaming Algorithms for Submodular Maximization with Matroid Constraints

TL;DR

This paper introduces improved multi-pass streaming algorithms for maximizing submodular functions under matroid constraints, achieving better approximation guarantees with fewer passes and less memory, applicable to both monotone and non-monotone cases.

Contribution

It presents the first multi-pass streaming algorithms with guarantees for general non-monotone submodular maximization under p-matchoid constraints, improving efficiency and approximation ratios.

Findings

Achieves a (p+1+ε)-approximation with O(p/ε) passes for monotone functions.

Provides a (p+1+γ̄+O(ε))-approximation for non-monotone functions with limited memory.

Algorithms are oblivious to ε and can be stopped at any pass, maintaining guarantees.

Abstract

We give improved multi-pass streaming algorithms for the problem of maximizing a monotone or arbitrary non-negative submodular function subject to a general $p$-matchoid constraint in the model in which elements of the ground set arrive one at a time in a stream. The family of constraints we consider generalizes both the intersection of $p$ arbitrary matroid constraints and $p$-uniform hypergraph matching. For monotone submodular functions, our algorithm attains a guarantee of $p+1+\varepsilon$ using $O(p/\varepsilon)$-passes and requires storing only $O(k)$ elements, where $k$ is the maximum size of feasible solution. This immediately gives an $O(1/\varepsilon)$-pass $(2+\varepsilon)$-approximation algorithms for monotone submodular maximization in a matroid and $(3+\varepsilon)$-approximation for monotone submodular matching. Our algorithm is oblivious to the choice $\varepsilon$ and can be stopped after any number of passes, delivering the appropriate guarantee. We extend our techniques to obtain the first multi-pass streaming algorithm for general, non-negative submodular functions subject to a $p$-matchoid constraint with a number of passes independent of the size of the ground set and $k$. We show that a randomized $O(p/\varepsilon)$-pass algorithm storing $O(p^3k\log(k)/\varepsilon^3)$ elements gives a $(p+1+\bar{\gamma}+O(\varepsilon))$-approximation, where $\bar{gamma}$ is the guarantee of the best-known offline algorithm for the same problem.